A Method to Reduce the Complexity of Computing the Weight Distribution of Polar Codes

The code spectrum of a linear code provides insight into its optimal performance. By leveraging the lower-triangular affine group (LTA) of decreasing monomial codes in conjunction with the one-variable descendant (ovd) relation, we introduce a novel subgroup of LTA that can identify additional cosets with identical weight distributions. By exploiting this algebraic structure, we demonstrate that the group action on a coset set is transitive. Our method advances previous research, reducing complexity by several times in many cases of code length $N=128$ and $N=256$.